Stationary solution for transient quantum hydrodynamics with bohmenian-type boundary conditions

We derive rigorously a set of boundary conditions for heterogenous devices using a description via the quantum hydrodynamic system provided by the Madelung transformations. In particular, we show that the generalized enthalpy should be constant at the interface between classical and quantum domains. This condition provides a set of boundary conditions, which we use to prove the existence and the uniqueness of regular steady solutions of the quantum hydrodynamic system. Finally, we analyse the linear stability of the system supplied with our boundary conditions and we test numerically our model on a toy device.